Transport in disordered graphene nanoribbons Ivar Martin (LANL) Yaroslav Blanter (Delft)
Weak disord.:
Strong disord.:
Outline
• Experiments on graphene nanoribbons GNR (IBM and Columbia, 2007) • Model of weakly disordered ribbons (solvable but artificial, somewhat) • Model for strongly disordered ribbons (interesting but hard) • Ballistic transport through metal-graphene-metal junctions – “contact-independent” conductance
Graphene in the grand scheme of things
Borrowed from Geim’s 2007 KITP presentation
Fabrication and field effect
K’ K
Geim and Novoselov, Nature Materials, 2007
: BZ
Ribbons
20 nm 30 nm 40 nm 50 nm 100 nm 200 nm
Chen, Lin, Rooks, and Avouris, cond-mat 2007
Han, Özyilmaz, Zhang, and Kim, cond-mat 2007
I(T) and current 1/f noise
e
− E g / 2T
?
Chen, Lin, Rooks, Avouris (IBM), cond-mat 2007
Width-dependence of gap, Eg(W)
Eg =
α W − W0
Han, Özyilmaz, Zhang, and Kim, cond-mat 2007
Questions to be answered
• • • • •
What causes activated behavior? Why at T= 4K conductivity is relatively high? [Is it a real semiconducting gap?] How can gap depend on the width?! What causes 1/f noise: background charge fluctuations or intrinsic fluctuations?
What is known: clean perfect ribbons
• zig-zag are gapless (edge modes) • armchairs: gapped or gapless W = 3N, 3N+2
W = 3N+1
Ezawa, PRB 2006; Brey-Fertig, PRB 2006
Zig-zag ribbons and zig-zag edges Zero-energy evanescent states in semi-infinite graphene: ⎛ 0 ⎜ ⎝ px − ip y
Edge:
⎛ e − k y x +ik y y ⎞ ⎜⎜ ⎟⎟ ⎝ 0 ⎠
px + ip y ⎞ ⎛ a ⎞ =0 ⎟ 0 ⎠ ⎜⎝ b ⎟⎠ and
⎛ 0 ⎞ ⎜⎜ k y x +ik y y ⎟⎟ ⎝e ⎠
• Large ky – strong localization • In ribbon edge states interact x
NAKADA, FUJITA, DRESSELHAUS, AND DRESSELHAUS, PRB 1996
unhappy
happy
Armchair ribbons
Ribbon crosssection
Simplest model of disorder variable-width armchair GNR
~W
~L
~L
~L
~L
Take L >> W Metal – level spacing t/L Insulator – gap t/W
Band structure and Wave-function segmentation!
Example (Eg = 2 x 0.17 t)
(Eg = 2 x 0.17 t)
Eg
E = -0.1 t
E = -0.2 t
E = -0.4 t
Effective model 1D hopping tij
εi-1
εi
εj
εi
εi+1
εi+2
WKB: tij ~ e− 2m*U r
U ~ t /W ,
massin insulator :
ε ins =
Δεi ~ t/Li
U2+
p2vF2
2 p =U + 2(U / vF2 )
thus m* ~ (tag2W )−1 and
−rij /W
tij ~ e
2U
Hopping transport
σ ∼e
− rij / W −|ε i −ε j |/ T
for a hop of length
n: rij = nLav and | ε i − ε j |~ (t / Lav ) / n
Minimize over n n > 1: VRH av n = 1: NNH Crossover at
Hopping transport regimes
av log σ
~1/T ~1/ T Lav t
L2av tW
1/T
Strong disorder case
E = -0.07 t
E = -0.09 t
E = -0.255 t
Still segmented but with different degree of localization!
Strong disorder case – schematic
States with E < t/W
Counting states:
per plaquette W × W dos 2D × (t / W ) × W 2 ~ 1 state / plaquette Which states contribute to conductivity?
only “good” fat states All states
Two possibilities
W 1
W
W 2
W2
Compare with expt
e
− E g / 2T
?
log σ NNH ~ 1/T
~ 1/ T
Chen, Lin, Rooks, Avouris (IBM), cond-mat 2007
W t
W 2 t
VRH
1/T
Not all low energy states matter!
How about Coulomb? Efros-Shklovskii: extra Coulomb cost, e2/r
r – length of hop
e 2 / tag ~ 1 Graphene fine-structure constant Thus: • For long-hops via “good” states Coulomb is irrelevant • For short hops via all low energy states can be relevant
Can be tested by placing a metallic gate to screen Coulomb
Metal-Graphene-Metal (MGM)
Transport through NGN structure ts
t'
tg
tg
t'
ts
Indep of β!
t '2 β= t g ts V qg = t g ag
Conclusions
• Simple model for disordered nanoribbons • Low temperature transport in ribbons can be in variable or nearest neighbor hopping regimes • Coulomb appears irrelevant for the hopping through good fat states • 1/f noise appears to be consistent with hopping conductivity • NGN structure, resonances, and contact-independent resistance
